6 problems

Spanning-set reduction conjecture. The spanning set can be reduced by removing every vector $\alpha(-\lambda)X(-\mu)v_\Lambda$ for which the partition $\mu$ has a sub-partition, no…

Let $\mathcal G(F)$ be a reductive $p$-adic group, let $\pi_{st}$ be a standard representation of $\mathcal G(F)$, and let $\pi$ be an irreducible subquotient of $\pi_{st}$. Assume…

Let $\mathcal G(F)$ be a quasi-split reductive $p$-adic group. Let $\mathcal P(F)=\mathcal M(F)\mathcal U_{\mathcal P}(F)$ be a parabolic subgroup, let $\tau$ be an irreducible tem…

Let $G$ be a quasi-split reductive $p$-adic group, let $\pi_{st}$ be a standard representation of $G$, and let $\pi$ be an irreducible subquotient of $\pi_{st}$. A representation i…

Monomial-basis conjecture. The displayed spanning set is a basis of $L(k\Lambda_0)$ for all $k\geq 1$ and $\ell\geq 1$. This extends the cases already known for type $A_1$ at all p…

Let $ngeq 2$ and $kgeq 2$. Consider the standard module $L(k\Lambda_0)$ for the affine Lie algebra of type $C_n^{(1)}$, with generators $X_{ab}(j)$ indexed by ${ab}\in B$ and…